#9945581

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,20,22,18)(19,25,27,23)(21,24,29,28)(26,30,32,31)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,47)(34,40,41,45)(35,42,43,48)(36,44,39,46)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852777);